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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.7.60a
Chapter 3, Problem 3.7.60a

Computer Explorations


Use a CAS to perform the following steps in Exercises 55–62.


a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.


xy³ + tan(x + y) = 1, P(π/4, 0)

Verified step by step guidance
1
Start by understanding the given equation: \(xy^3 + \tan(x + y) = 1\). This is an implicit equation involving both x and y.
Use a Computer Algebra System (CAS) to plot the equation. Input the equation into the implicit plotter feature of the CAS to visualize the curve represented by the equation.
To verify if the point P(\(\frac{\pi}{4}, 0\)) satisfies the equation, substitute \(x = \frac{\pi}{4}\) and \(y = 0\) into the equation.
Calculate the left-hand side of the equation with the substituted values: \(\left(\frac{\pi}{4}\right)(0)^3 + \tan\left(\frac{\pi}{4} + 0\right)\).
Check if the calculated value from the previous step equals 1, which is the right-hand side of the equation. If it does, then the point P satisfies the equation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Plotting

Implicit plotting involves graphing equations where the relationship between variables is not explicitly solved for one variable in terms of the others. In this case, the equation xy³ + tan(x + y) = 1 is plotted using a Computer Algebra System (CAS) to visualize solutions that satisfy the equation, including checking specific points like P(π/4, 0).
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Finding The Implicit Derivative

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is software designed to perform symbolic mathematical computations. It can handle tasks such as algebraic simplifications, solving equations, and plotting graphs. In this exercise, a CAS is used to plot the implicit equation and verify if the point P satisfies the equation, showcasing its utility in handling complex mathematical operations.
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Verification of Points on Curves

Verifying if a point lies on a curve involves substituting the point's coordinates into the equation and checking if the equation holds true. For the equation xy³ + tan(x + y) = 1, substituting P(π/4, 0) checks if the left-hand side equals 1, confirming whether P is a solution to the equation. This process is crucial for understanding the relationship between the equation and specific points.
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Summary of Curve Sketching
Related Practice
Textbook Question

Finding Derivative Values


In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.


f(u) = u + 1/cos²u, u = g(x) = πx, x = 1/4

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Textbook Question

Assume that f'(3) = −1, g'(2) = 5, g(2) = 3, and y = f(g(x)). What is y' at x = 2?

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Textbook Question

In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.


y = (4 − 3x)⁹

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Textbook Question

The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s


a. surface area?

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Textbook Question

Find y⁽⁴⁾ = d⁴y/dx⁴ if:

a. y = −2 sin x

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Textbook Question

Differentiability and Continuity on an Interval


Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be


a. differentiable?


Give reasons for your answers.


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